During the history of research on multiple maternities, Hellin's law has played a central role as a rule of thumb. It is mathematically simple and approximately correct, but shows discrepancies that are difficult to explain or to eliminate. It has been mathematically proven that Hellin's law does not hold as a general rule. Varying improvements to this law have been proposed. In this paper, we consider how Hellin's law can be used and tested in statistical analyses of the rates of multiple maternities. Such studies can never confirm the law, but only identify errors too large to be characterized as random. It is of particular interest to determine why the rates of higher multiple maternities are sometimes too high or too low when Hellin's law is used as a benchmark. Excesses of triplet and quadruplet maternities are particularly unexpected and challenging. Our analyses of triplet and quadruplet rates indicated that triplet rates are closer to Hellin's law than quadruplet rates. According to our analyses of the twinning rate and the transformed triplet rate and quadruplet rate for Sweden (1751–2000), both triplet and quadruplet rates showed excesses after the 1960s. This is mainly caused by artificial fertility-enhancing reproduction technologies. Regression analyses of twinning and triplet rates yield rather good fits, but deficiencies in the triplet rates are commonly present. We introduced measures of concordance between triplet rates with Hellin's law. According to these measures, historic data showed deficiencies in triplet rates, but recent data revealed excesses, especially found among older mothers. The excesses obtained are in good agreement with other studies of recent data.
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